Derivadas Imposibles

        Derivadas Imposibles

∂   Derivada "Imposible" 1     $$ y=\sin^{-1}\left(\frac{\sqrt{a-b}~\sin(\frac{x}{2})}{\sqrt{a+b~\cos(x)}}\right) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{\sqrt{a^2-b^2}}{2(a+b~\cos(x))} $$

∂   Derivada "Imposible" 2     $$ y={\frac{1}{\sqrt{b^2-a^2}}~{\sin^{-1}\left(\frac{\sqrt{b^2-a^2}~\sinh(x)}{a+b~\cosh(x)}\right)}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{a+b~\cosh(x)} $$

∂   Derivada "Imposible" 3     $$ y={\frac{1}{\sin(a)}}~{\cosh^{-1}\left(\frac{1+\cos(a)\cos(x)}{\cos(a)\cos(x)}\right)} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{\cos(a)+\cos(x)} $$

∂   Derivada "Imposible" 4     $$ y=\tan^{-1}\left(\sinh(x^2)\right) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=(2x)~\textrm{sech}(x^2) $$

∂   Derivada "Imposible" 5     $$ y=x^{x^{x}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=(x^x)^{x+1}~\left(\frac{1}{x}+\ln(x)+\ln^2(x)\right) $$

∂   Derivada "Imposible" 6     $$ y=\left(\cos(x)\right)^{\sin(x)} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\left(\cos(x)\right)^{\sin(x)}~\left(\cos(x)~\ln(\cos(x))-\sin(x)~\tan(x)\right) $$

∂   Derivada "Imposible" 7     $$ y=\sqrt[3]{\frac{(x+1)^2~(x+3)}{(x+2)~(x+4)^2}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{3}~\sqrt[3]{\frac{(x+1)^2~(x+3)}{(x+2)~(x+4)^2}}~\left(\frac{2}{x+1}+\frac{1}{x+3}-\frac{1}{x+2}-\frac{2}{x+4}\right)$$

∂   Derivada "Imposible" 8     $$ y=\frac{e^{a~\tan^{-1}(x)}~(a+x)}{(1+a^2)~\sqrt{1+x^2}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{e^{a~\tan^{-1}(x)}}{(1+x^2)^\frac{3}{2}} $$

∂   Derivada "Imposible" 9     $$ y=\frac{1}{\sqrt{5}}~\ln\left(\frac{\tan(\frac{x}{2})+\sqrt{5}}{\tan(\frac{x}{2})-\sqrt{5}}\right) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{2+3\cos(x)} $$

∂   Derivada "Imposible" 10     $$ y=\frac{1}{n}~\log_e\left(\frac{\sqrt{1+e^{nx}}-1}{\sqrt{1+e^{nx}}+1}\right) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{\sqrt{1+e^{nx}}} $$

∂   Derivada "Imposible" 11     $$ y=\frac{1}{ab}~\tan^{-1}\left(\frac{b}{a}\tan(x)\right) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1}{a^2\cos^2(x)+b^2\sin^2(x)} $$

∂   Derivada "Imposible" 12     $$ y=\frac{2bx^2-a}{x^3}~\sqrt{a+bx^2} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{3a^2}{x^4\sqrt{a+bx^2}} $$

∂   Derivada "Imposible" 13     $$ x^ny^m=(x+y)^{m+n} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{y}{x} $$

∂   Derivada "Imposible" 14     $$ y=\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{\sqrt{x^2-1}-x}{\sqrt{x^2-1}} $$

∂   Derivada "Imposible" 15     $$ y=|x| \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{x}{|x|} $$

∂   Derivada "Imposible" 16     $$ y=\sqrt[3]{|x|+x} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{\sqrt[3]{|x|+x}}{3|x|} $$

∂   Derivada "Imposible" 17     $$ y=\sqrt[4]{\frac{x^3+1}{x^3-1}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{-3x^4}{2(x^3-1)\sqrt[4]{(x^3-1)(x^3+1)^3}} $$

∂   Derivada "Imposible" 18     $$ x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{x^{\frac{4}{3}}+y^{\frac{2}{3}}~x^{\frac{2}{3}}}{3~y^{\frac{1}{3}}~x^2} $$

∂   Derivada "Imposible" 19     $$ e^x=\log_{\sin(y)}(x+y) \hspace{0.8 cm} \longrightarrow \hspace{0.8 cm} \frac{dy}{dx}=\frac{1-e^x~\ln(\sin(y))~(x+y)}{\cot(y)(x+y)~e^x-1} $$